Time Integration

Various implementations of the method of lines to progress through time. The goal is to implement the code in python and not rely on existing solvers.

Bram Lagerweij COHMAS Mechanical Engineering KAUST 2021

time_integral.backwardEuler(func, u, dt, t_end, args=())

Itterate a through time with the backward Eurler method.

The backward Euler method predicts the field of our function based upon information of the previous timestep only. Imagine that we are at timestep \(n\) and want to predict our field at timestep \(u^{(n+1)}\). Now a backward finite difference approximation used the time derivative of the next timestep, wich is not yet known:

\[u^{(n+1)}_t = \frac{ -u^{(n)} + u^{(n+1)} }{dt}\]

That is we can predict our field in the future timestep as:

\[u^{(n+1)} = u^{(n)} + dt\, u^{(n+1)}_t\]

It is important to notic that there is a term with an unknown, as that is at time step :math:`n+1’ on both sides of the equation. Our time derivative is obtained with an approximation equation:

\[u_t = K u + b\]

where matrix \(K\) and vector \(b\) stem from approximations of our spatial derivatives defined by the functien provided to func. This results in:

\[u^{(n+1)} = u^{(n)} + dt\, ( K u^{(n+1)} + b )\]

Now we rewrite it into a system of equations where we find all unknowns on the left hand side and all knownn on the right hand side.

\[(I - dt\,K)\, u^{(n+1)} = u^{(n)} + dt\,b\]

Parameters

• func (callable) – The time derivative of the pde to be solved such that \(u_t = K\,u + b\).

• u (array_like) – The field at the start \(u(t=0)\).

• dt (float) – The size of the time step.

• t_end (float) – Time at termination.

• args (tuple, optional) – The parameters into the PDE approximation. Defealts to an empty tuple.

Returns

The function for all time steps.

Return type

array_like

time_integral.forwardEuler(func, u, dt, t_end, args=())

Itterate a through time with the forward Eurler method.

The backward Euler method predicts the field of our function based upon information of the previous timestep only. Imagine that we are at timestep \(n\) and want to predict our field at timestep \(u^{(n+1)}\). Now a forward finite difference approximation is used:

\[u^{(n)}_t = \frac{-u^{(n)} + u^{(n+1)} }{dt}\]

That is we can predict our field in the future timestep as:

\[u^{(n+1)} = u^{(n)} + dt\, u^{(n)}_t\]

Our time derivative at the current timestep, \(u^{(n)}_t\) is obtained with:

\[u_t = K u + b\]

where matrix \(K\) and vector \(b\) stem from approximations of our spatial derivatives defined by the functien provided to func. Resulting in the following update scheme:

\[u^{(n+1)} = u^{(n)} + dt\, (K u^{(n)} + b)\]

most important of all is to see that everything on the right hand side is exactly known. Thus the updated field can be calculated directly.

Parameters

• func (callable) – The time derivative of the pde to be solved such that \(u_t = K\,u + b\).

• u (array_like) – The field at the start \(u(t=0)\).

• dt (float) – The size of the step.

• t_end (float) – Time at termination.

• args (tuple, optional) – The parameters into the PDE approximation. Defealts to an empty tuple.

Returns

The function for all time steps.

Return type

array_like